Question

find the variance of the data. 87, 94, 103, 84, 112, 90 \\( \overline { x } = 95 \\) variance \\( ( \sigma ^ { 2 } ) = \\)

Explanation:

Step1: Calculate the squared differences

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\).

• For \(x_1 = 87\): \((87 - 95)^2=(-8)^2 = 64\)
• For \(x_2 = 94\): \((94 - 95)^2=(-1)^2 = 1\)
• For \(x_3 = 103\): \((103 - 95)^2=(8)^2 = 64\)
• For \(x_4 = 84\): \((84 - 95)^2=(-11)^2 = 121\)
• For \(x_5 = 112\): \((112 - 95)^2=(17)^2 = 289\)
• For \(x_6 = 90\): \((90 - 95)^2=(-5)^2 = 25\)

Step2: Sum the squared differences

\(\sum_{i = 1}^{6}(x_i-\bar{x})^2=64 + 1+64+121+289+25\)
\(=564\)

Step3: Divide by the number of data points \(n\)

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}\). Here \(n = 6\)
\(\sigma^{2}=\frac{564}{6}\)

Answer:

\(94\)